Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance.

Discrete systems of infinitely long polarizable line dipoles are considered in the quasistatic limit, interacting with a two-dimensional cloaking system consisting of a hollow plasmonic cylindrical shell. A numerical procedure is described for accurately calculating electromagnetic fields arising in the quasistatic limit, for the case when the relative permittivity of the cloaking shell has a very small imaginary part. Animations are given which illustrate cloaking of discrete systems, both for the case of induced dipoles and induced quadrupoles on the interacting particles. The simulations clarify the physical mechanism for the cloaking.

Conductance of photons in disordered photonic crystals.

The conductance of photons in two-dimensional disordered photonic crystals is calculated using an exact multipole-plane wave method that includes all multiple scattering processes. Conductance fluctuations, the universal nature of which has been established for electrons in the diffusive regime, are studied for photons, in both principal polarizations and for varying disorder. Our simulations show that universal conductance fluctuations can be observed in H(||) (TE) polarization for weak and intermediate disorder while, for E(||) (TM) polarization, we show that the conductance variance is essentially independent of sample size but strongly dependent on disorder. The probability distribution of the conductance is also calculated in the diffusive and localized regimes, and also at their transition, for which the distributions for both polarizations are seen to be very similar.

Density of states functions for photonic crystals.

We discuss density of states functions for photonic crystals, in the context of the two-dimensional problem for arrays of cylinders of arbitrary cross section. We introduce the mutual density of states (MDOS), and show that this function can be used to calculate both the local density of states (LDOS), which gives position information for emission of radiation from photonic crystals, and the spectral density of states (SDOS), which gives angular information. We establish the connection between MDOS, LDOS, SDOS and the conventional density of states, which depends only on frequency. We relate all four functions to the band structure and propagating states within the crystal, and give numerical examples of the relation between band structure and density of states functions.

Cylinder gratings in conical incidence with applications to woodpile structures.

We use our previous formulation for cylinder gratings in conical incidence to discuss the photonic band gap properties of woodpile structures. We study scattering matrices and Bloch modes of the woodpile, and use these to investigate the dependence of the optical properties on the number of layers. We give data on reflectance, transmittance and absorptance of metallic woodpiles as a function of wavelength and number of layers, using both the measured optical constants of tungsten and using a perfect conductivity idealization to characterize the metal. For semi-infinite metallic woodpiles, we show that polarization of the incident field is important, highlighting the role played by surface effects as opposed to lattice effects.

Diffusion and anomalous diffusion of light in two-dimensional photonic crystals.

The transport properties of electromagnetic waves in disordered, finite, two-dimensional photonic crystals composed of circular cylinders are considered. Transport parameters such as the transport and scattering mean free paths and the transport velocity are calculated, for the case where the electromagnetic radiation has its electric field along the cylinder axes. The range of the parameters in which the diffusion process can take place is specified. It is shown that the transport velocity upsilon(E) can be as much as 10(8) times less than its free space value, while just outside the cluster upsilon(E) can be 0.3c. The effects of weak and strong disorders on the transport velocity are investigated. Different regimes of the wave transport-ordered propagation, diffusion, and anomalous diffusion-are demonstrated, and it is inferred that Anderson localization is incipient in the latter regime. Exact numerical calculations from the Helmholtz equation are shown to be in good agreement with the diffusion approximation.

Cylinder gratings in conical incidence with applications to modes of air-cored photonic crystal fibers.

We develop a formulation for cylinder gratings in conical incidence, using a multipole method. The theory, and its numerical implementation, is applied to two-dimensional photonic crystals consisting of a stack of one-dimensional gratings, each characterized by its plane wave scattering matrix. These matrices are used in combination with Bloch's theorem to determine the band structure of the photonic crystal from the solution of an eigenvalue problem. We show that the theory is well adapted to the difficult task of locating the complete band gaps needed to support air-guided modes in microstructured optical fibers, that is, optical fibers in which the confinement of light in a central air hole is achieved by photonic band-gap effects in a periodic cladding comprising a lattice of air holes in a glass matrix.

Photonic band structure calculations using scattering matrices.

We consider band structure calculations of two-dimensional photonic crystals treated as stacks of one-dimensional gratings. The gratings are characterized by their plane wave scattering matrices, the calculation of which is well established. These matrices are then used in combination with Bloch's theorem to determine the band structure of a photonic crystal from the solution of an eigenvalue problem. Computationally beneficial simplifications of the eigenproblem for symmetric lattices are derived, the structure of eigenvalue spectrum is classified, and, at long wavelengths, simple expressions for the positions of the band gaps are deduced. Closed form expressions for the reflection and transmission scattering matrices of finite stacks of gratings are established. A new, fundamental quantity, the reflection scattering matrix, in the limit in which the stack fills a half space, is derived and is used to deduce the effective dielectric constant of the crystal in the long wavelength limit.

Photonic engineering. Aphrodite's iridescence.

The most intense colours displayed in nature result from either multilayer reflectors or linear diffraction gratings. Here we investigate the spectacular iridescence of a spine (notoseta) from the sea mouse Aphrodita sp. (Polychaeta: Aphroditidae). The spine normally appears to be deep red in colour, but when light is incident perpendicular to the axis of the spine, different colours are seen as stripes running parallel to the axis of the spine; over a range of smaller incident angles, the complete visible spectrum is reflected with a reflectivity of 100% to the human eye. The simple structure responsible for this effect is a remarkable example of photonic engineering by a living organism.

Two-dimensional Green's function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length.

Using the exact theory of multipole expansions, we construct the two-dimensional Green's function for photonic crystals, consisting of a finite number of circular cylinders of infinite length. From this Green's function, we compute the local density of states (LDOS), showing how the photonic crystal affects the radiation properties of an infinite fluorescent line source embedded in it. For frequencies within the photonic band gap of the infinite crystal, the LDOS decreases exponentially inside the crystal; within the bands, we find "hot" and "cold" spots. Our method can be extended to three dimensions as well as to treating disorder and represents an important and efficient tool for the design of photonic crystal devices.

Two-dimensional local density of states in two-dimensional photonic crystals.

We calculate the two-dimensional local density of states (LDOS) for two-dimensional photonic crystals composed of a finite cluster of circular cylinders of infinite length. The LDOS determines the dynamics of radiation sources embedded in a photonic crystal. We show that the LDOS decreases exponentially inside the crystal for frequencies within a photonic band gap of the associated infinite array and demonstrate that there exist ;;hot' and ;;cold' spots inside the cluster even for wavelengths inside a gap, and also for wavelengths corresponding to pass bands. For long wavelengths the LDOS exhibits oscillatory behavior in which the local density of states can be more than 30 times higher than the vacuum level.

Microstructured polymer optical fibre.

The first microstructured polymer optical fibre is described. Both experimental and theoretical evidence is presented to establish that the fibre is effectively single moded at optical wavelengths. Polymer-based microstructured optical fibres offer key advantages over both conventional polymer optical fibres and glass microstructured fibres. The low-cost manufacturability and the chemical flexibility of the polymers provide great potential for applications in data communication networks and for the development of a range of new polymer-based fibre-optic components.

Ring structures in microstructured polymer optical fibres.

Recent developments in polymer microstructured optical fibres allow for the realisation of microstructures in fibres that would be problematic to fabricate using glass-based capillary stacking. We present one class of such structures, where the holes lie on circular rings. A fibre of this type is fabricated and shown to be single moded for relatively long lengths of fibre, whereas shorter lengths are multimoded. An average index model for these fibres is developed. Comparison of its predictions to the calculated properties of the exact structure indicates that the ring structures emulate homogeneous rings of lower refractive index resulting in the ring structured fibres behaving approximately as cylindrically layered fibres.

Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method.

We present a formulation for wave propagation and scattering through stacked gratings comprising metallic and dielectric cylinders. By modeling a photonic crystal as a grating stack of this type, we thus formulate an efficient and accurate method for photonic crystal calculations that allows us to calculate reflection and transmission matrices. The stack may contain an arbitrary number of gratings, provided that each has a common period. The formulation uses a Green's function approach based on lattice sums to obtain the scattering matrices of each layer, and it couples these layers through recurrence relations. In a companion paper [J. Opt Soc. Am. A 17, 2177 (2000)] we discuss the numerical implementation of the method and give a comprehensive treatment of its conservation properties.

Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part II. Properties and implementation.

A numerical implementation and generalized conservation properties of a formulation for calculating wave propagation through stacked gratings comprising metallic and dielectric cylinders are presented. The basic formulation of the method was given in a companion paper [J. Opt. Soc. Am. A. 17, 2165 (2000)]. Here, details of the numerical implementation of the method are discussed and are illustrated for the ensemble average of a strongly scattering structure with refractive index and radius disorder. Also presented are a comprehensive treatment of energy conservation and generalized phase relations, as well as reciprocity.

Addition formulas and the Rayleigh identity for arrays of elliptical cylinders.

We apply the Rayleigh method to solve the problem where a uniform electrostatic field is imposed upon a rectangular array of elliptical cylinders embedded in a matrix of unit dielectric constant. This new formulation overcomes geometric restrictions inherent in previous methods and is shown in principle and in various examples to converge for all possible geometries of the array and inclusion. Also presented are forms of both the interior and exterior addition formulas for harmonic functions in elliptical coordinates that possess optimal regions of convergence.

Effects of disorder on wave propagation in two-dimensional photonic crystals.

The electromagnetic transmittance of disordered two-dimensional photonic crystals composed of circular cylinders is investigated as a function of wavelength and polarization. At short wavelengths, the transmittance shows a band structure similar to that found in the optical absorption spectrum of amorphous semiconductors, with impurity states increasingly appearing on the long wavelength side of the band gaps as the degree of disorder is increased. In the long-wavelength limit, Anderson localization of waves is found, provided that the wavelength is not so large that the random photonic crystal can be viewed as homogeneous. The localization properties in this regime are studied and an analytic expression for the dependence of the localization length on wavelength is derived. In the limit of extremely long wavelengths, the system homogenizes and can be replaced by an equivalent one with uniform effective refractive index, whose form is derived for both polarizations. Analysis of the crossover between localization and homogenization is also presented.

Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders.

A method is developed to calculate electromagnetic properties of arrays of metallic and dielectric cylinders. It incorporates and exploits cylindrical boundary conditions and Rayleigh identities for efficient, high-accuracy calculation of scattering off individual layers that are stacked into arrays using scattering matrices. The method enables absorption, dispersion, and randomness to be incorporated efficiently, and reproduces known results with vastly improved speed and accuracy. It is used to demonstrate existence of states introduced into photonic band gaps of a dielectric array by disorder, and anomalous absorption behavior in arrays of aluminum cylinders.